The stable Bernstein center and test functions for Shimura varieties

Haines, Thomas J.

doi:10.1017/cbo9781107297524.005

Cited by 52 publications

(103 citation statements)

References 60 publications

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“…which was proved in [H13,Remark 11.12.2]. In fact this is done by interpreting (8.2), via the Kottwitz isomorphism, as the natural map…”

Section: Review Of Transfer Homomorphismsmentioning

confidence: 99%

“…We now recall the definition of a normalized version of t A * ,A , for which we need to refine the choice of ψ 0 ∈ Ψ M somewhat. Following [H13,Lemma 11.12.4], given the choice of P ⊃ M and B * ⊃ T * used to define Ψ M , choose any F un -rational ψ 0 ∈ Ψ M and define a morphism of affine algebraic varieties…”

Section: Review Of Transfer Homomorphismsmentioning

confidence: 99%

“…As explained in [H13,Def. 11.12.5],t is independent of the choices for A, A * , and B * ⊃ T * , and is a completely canonical homomorphism.…”

Section: Review Of Transfer Homomorphismsmentioning

confidence: 99%

“…So for example an Iwahori subgroup here is somewhat smaller than the "naive" notion that sometimes appears in the literature under the same name, and therefore the Iwahori-Hecke algebra and its center are slightly larger (cf. [H09c] and [H13,Appendix]). …”

Section: Introductionmentioning

confidence: 99%

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On Satake Parameters for Representations with Parahoric Fixed Vectors

Haines

2015

Int Math Res Notices

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Abstract. This article, a continuation of [HRo], constructs the Satake parameter for any irreducible smooth J-spherical representation of a p-adic group, where J is any parahoric subgroup. This parametrizes such representations when J is a special maximal parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in formulating a geometric Satake isomorphism for such groups over local function fields.

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“…which was proved in [H13,Remark 11.12.2]. In fact this is done by interpreting (8.2), via the Kottwitz isomorphism, as the natural map…”

Section: Review Of Transfer Homomorphismsmentioning

confidence: 99%

Section: Review Of Transfer Homomorphismsmentioning

confidence: 99%

“…As explained in [H13,Def. 11.12.5],t is independent of the choices for A, A * , and B * ⊃ T * , and is a completely canonical homomorphism.…”

Section: Review Of Transfer Homomorphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Satake Parameters for Representations with Parahoric Fixed Vectors

Haines

2015

Int Math Res Notices

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“…Let I F be the inertia subgroup of W F and let Frob F 2 W F be a Frobenius element, so that [Hai,(3.3.2)] there are natural isomorphisms…”

Section: Towards a Galois Analogue Of The Bernstein Theorymentioning

confidence: 99%

Conjectures about 𝑝-adic groups and their noncommutative geometry

Aubert¹,

Baum²,

Plymen

et al. 2017

Around Langlands Correspondences

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Abstract. Let G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G.At the heart of these conjectures are statements about the geometric structure of Bernstein components for G, both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the BaumConnes conjecture for G. In particular, we present a strategy to reduce the local Langlands correspondence for irreducible G-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups. Contents

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Graded Hecke Algebras for Disconnected Reductive Groups

2018

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We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G.We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data (G, M, L) and they are closely related to Langlands parameters.Our main motivation for considering these graded Hecke algebras is that the space of irreducible H-representations is canonically in bijection with a certain set of "logarithms" of enhanced L-parameters. Therefore we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper. Erratum. Theorem 3.4 and Proposition 3.22 were not entirely correct as stated. This is repaired in a new appendix.

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The stable Bernstein center and test functions for Shimura varieties

Cited by 52 publications

References 60 publications

On Satake Parameters for Representations with Parahoric Fixed Vectors

On Satake Parameters for Representations with Parahoric Fixed Vectors

Conjectures about 𝑝-adic groups and their noncommutative geometry

Graded Hecke Algebras for Disconnected Reductive Groups

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